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A recursive construction for projective Reed-Muller codes

Rodrigo San-José

Abstract

We give a recursive construction for projective Reed-Muller codes in terms of affine Reed-Muller codes and projective Reed-Muller codes in fewer variables. From this construction, we obtain the dimension of the subfield subcodes of projective Reed-Muller codes for some particular degrees that give codes with good parameters. Moreover, from this recursive construction we derive a lower bound for the generalized Hamming weights of projective Reed-Muller codes which is sharp in most of the cases we have checked.

A recursive construction for projective Reed-Muller codes

Abstract

We give a recursive construction for projective Reed-Muller codes in terms of affine Reed-Muller codes and projective Reed-Muller codes in fewer variables. From this construction, we obtain the dimension of the subfield subcodes of projective Reed-Muller codes for some particular degrees that give codes with good parameters. Moreover, from this recursive construction we derive a lower bound for the generalized Hamming weights of projective Reed-Muller codes which is sharp in most of the cases we have checked.
Paper Structure (9 sections, 18 theorems, 71 equations, 8 tables)

This paper contains 9 sections, 18 theorems, 71 equations, 8 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A recursive construction for projective Reed-Muller codes
  4. Subfield subcodes of projective Reed-Muller codes
  5. Examples
  6. A bound for the generalized Hamming weights of projective Reed-Muller codes
  7. A bound for the generalized Hamming weights of projective Reed-Muller codes over $\mathbb{P}^2$
  8. A bound for the generalized Hamming weights of the subfield subcodes of projective Reed-Muller codes
  9. Examples

Key Result

Theorem 2.1

The projective Reed-Muller code $\mathop{\mathrm{PRM}}\nolimits_d(q^s,m)$, $1\leq d\leq m(q^s-1)$, is an $[n,k]$-code with For the minimum distance, we have

Theorems & Definitions (39)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Corollary 3.3
  • proof
  • Corollary 4.1
  • proof
  • ...and 29 more